Karoubi–Villamayor K-theory

Summary

Karoubi–Villamayor K-theory assigns to a ring $R$ groups $KV_n(R)$, $n \ge 1$, built from invertible matrices over the rings of polynomial functions on algebraic simplices. By construction it is homotopy invariant: $KV_n(R[t]) \cong KV_n(R)$. It agrees with Quillen's algebraic K-theory on regular Noetherian rings, but differs in general. It was one of the first definitions of higher K-groups, and it shows what part of K-theory survives when polynomial deformations are made trivial.